Subgroups — #2932

Order 128.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	2	yes	maximal, complemented
2	#532	C₂	1	1	yes	maximal, characteristic
3	#435	C₂	1	8	yes	maximal, complemented
4	#423	C₂	1	1	yes	maximal, characteristic, complemented
5	#408	C₂	1	2	yes	maximal, complemented
6	C₂×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41	C₂²	1	2	yes	—
8	#131	—	2	8	no	—
9	C₂×M₁₆	C₂²	1	2	yes	—
10	C₂×M₁₆	C₂²	1	4	yes	—
11	C₂²×C₈	C₂²	1	4	yes	—
12	C₂²×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈	C₄	1	1	yes	characteristic
15	#127	C₄	1	1	yes	characteristic
16	#124	C₄	1	2	yes	—
17	#121	C₄	1	4	yes	—
18	C₄×D₄	C₂²	1	4	yes	—
19	C₂×C₄²	C₂²	1	1	yes	characteristic
20	C₄⋊C₈	C₂²	1	4	yes	—
21	C₂²⋊C₈	C₂²	1	8	yes	—
22	#97	C₂²	1	4	yes	—
23	#41	—	2	4	no	—
24	#41	—	2	4	no	—
25	C₂×Q₈	C₂×C₄	1	2	yes	—
26	C₂×D₄	C₂×C₄	1	4	yes	—
27	C₂×D₄	C₂×C₄	1	2	yes	—
28	C₂×D₄	C₂×C₄	1	4	yes	—
29	C₂²×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₄	C₂³	1	4	yes	—
31	C₂²×C₄	C₂³	1	1	yes	characteristic
32	M₁₆	—	2	4	no	—
33	M₁₆	—	2	8	no	—
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	—	2	8	no	—
36	C₂×C₈	D₄	1	4	yes	complemented
37	C₂×C₈	C₂³	1	2	yes	—
38	C₂×C₈	C₂³	1	4	yes	—
39	C₂×C₈	C₂³	1	2	yes	—
40	C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄	C₂×C₄	1	8	yes	—
42	C₄²	C₂×C₄	1	1	yes	characteristic
43	C₄²	C₂×C₄	1	1	yes	characteristic
44	C₄²	C₂³	1	2	yes	—
45	C₂³	C₂²×C₄	1	4	yes	—
46	C₂³	C₂²×C₄	1	1	yes	characteristic
47	Q₈	—	2	4	no	—
48	D₄	—	2	8	no	—
49	D₄	—	2	4	no	—
50	D₄	—	4	4	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₂²×C₄	1	2	yes	—
55	C₂×C₄	C₂²×C₄	1	2	yes	—
56	C₂×C₄	C₂²×C₄	1	4	yes	—
57	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄	C₂×D₄	1	2	yes	—
59	C₂×C₄	#41	1	2	yes	—
60	C₂×C₄	C₂⁴	1	1	yes	characteristic
61	C₈	—	2	4	no	—
62	C₈	—	2	2	no	—
63	C₈	C₂×D₄	1	4	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	4	no	—
66	C₂²	—	2	8	no	—
67	C₂²	C₄×D₄	1	2	yes	—
68	C₂²	C₂³×C₄	1	1	yes	characteristic
69	C₄	—	2	4	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	C₄×D₄	1	2	yes	—
73	C₄	C₂²×D₄	1	1	yes	characteristic
74	C₄	C₂×#41	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	—	4	4	no	—
77	C₂	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂	#568	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#131	C₂	semidirect
3. #435	C₂	semidirect
4. #423	C₂	semidirect
5. #408	C₂	semidirect
6. C₂×#97	C₂	semidirect
36. C₂×C₈	D₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#131	2
#532	2
#435	2
#423	2
#408	2
C₂×#97	2